Strategy-Proof Classification
Reshef Meir
School of Computer Science and
Engineering, Hebrew University
A joint work with Ariel. D. Procaccia and Jeffrey S. Rosenschein
Strategy-Proof Classification
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An Example of Strategic Labels in Classification
Motivation
Our Model
Previous work (positive results)
(~12 minutes)
• An impossibility theorem
• More results (if there is time)
Introduction
Motivation
Model
Results
Strategic labeling: an example
ERM
5 errors
Introduction
Motivation
Model
Results
There is a better
classifier!
(for me…)
Introduction
Motivation
Model
Results
If I will only
change the
labels…
2+4 = 6 errors
Introduction
Motivation
Model
Results
Classification
The Supervised Classification problem:
– Input: a set of labeled data points {(xi,yi)}i=1..m
– output: a classifier c from some predefined
concept class C ( functions of the form f : X{-,+} )
– We usually want c to classify correctly not just the
sample, but to generalize well, i.e .to minimize
R(c) ≡ E(x,y)~D[ c(x)≠y ]
the expected number of errors w.r.t. the distribution D
Introduction
Motivation
Model
Results
Classification (cont.)
• A common approach is to return the ERM, i.e.
the concept in C that is the best w.r.t. the given
samples (has the lowest number of errors)
• Generalizes well under some assumptions on
the concept class C
With multiple experts, we can’t trust our ERM!
Introduction
Motivation
Model
Results
Where do we find “experts” with
incentives?
Example 1: A firm learning purchase patterns
– Information gathered from local retailers
– The resulting policy affects them
– “the best policy, is the policy that fits my pattern”
Introduction
Motivation
Model
Results
Example 2: Internet polls / expert systems
Users
Reported Dataset
Classifier
Classification
Algorithm
Introduction
Motivation
Model
Results
Related work
• A study of SP mechanisms in Regression learning
– O. Dekel, F. Fischer and A. D. Procaccia, Incentive Compatible Regression Learning,
SODA 2008
• No SP mechanisms for Clustering
– J. Perote-Peña and J. Perote. The impossibility of strategy-proof clustering,
Economics Bulletin, 2003
Introduction
Motivation
Model
Results
A problem instance is defined by
• Set of agents I = {1,...,n}
• A partial dataset for each agent i  I,
Xi = {xi1,...,xi,m(i)}  X
• For each xikXi agent i has a label yik{,}
– Each pair sik=xik,yik is an example
– All examples of a single agent compose the labeled dataset
Si = {si1,...,si,m(i)}
• The joint dataset S= S1 , S2 ,…, Sn is our input
– m=|S|
• We denote the dataset with the reported labels by S’
Introduction
Motivation
Model
Results
Input: Example
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Agent 1
Agent 2
Agent 3
X1  Xm1
Y1  {-,+}m1
X2  Xm2
Y2  {-,+}m2
X3  Xm3
Y3  {-,+}m3
S = S1, S2,…, Sn = (X1,Y1),…, (Xn,Yn)
Introduction
Motivation
Model
Results
Incentives and Mechanisms
• A Mechanism M receives a labeled dataset S’
and outputs c  C
• Private risk of i: Ri(c,S) = |{k: c(xik)  yik}| / mi
• Global risk: R(c,S) = |{i,k: c(xik)  yik}| / m
• We allow non-deterministic mechanisms
– The outcome is a random variable
– Measure the expected risk
Introduction
Motivation
Model
Results
ERM
We compare the outcome of M to the ERM:
c* = ERM(S) = argmin(R(c),S)
cC
r* = R(c*,S)
Can our mechanism
simply compute and
return the ERM?
Introduction
Motivation
Model
Results
Requirements
MOST
1. Good approximation:
IMPORTANT
S R(M(S),S) ≤ β∙r*
SLIDE
2. Strategy-Proofness (SP):
i,S,Si‘ Ri(M(S-i , Si‘),S) ≥ Ri(M(S),S)
• ERM(S) is 1-approximating but not SP
• ERM(S1) is SP but gives bad approximation
Introduction
Motivation
Model
Results
Restricted settings
• A very small concept class: |C| = 2
– There is a deterministic SP mechanism that
obtains a 3-approximation ratio
– This bound is tight
– Randomization can improve the bound to 2
R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification under Constant
Hypotheses: A Tale of Two Functions, AAAI 2008
Introduction
Motivation
Model
Results
Restricted settings (cont.)
• Agents with similar interests:
– There is a randomized SP 3-approximation
mechanism (works for any class C)
Agent 1
Agent 2
Agent 3
R. Meir, A. D. Procaccia and J. S. Rosenschein, Incentive Compatible Classification with Shared
Inputs, IJCAI 2009.
Introduction
Motivation
Model
Results
But not everything shines 
• Without restrictions on the input, we cannot
guarantee a constant approximation ratio
Our main result:
Theorem: There is a concept class C, for which
there are no deterministic SP mechanisms
with o(m)-approximation ratio
Introduction
Motivation
Model
Results
Deterministic lower bound
Proof idea:
– First construct a classification problem that is
equivalent to a voting problem with 3 candidates
– Then use the Gibbard-Satterthwaite theorem to
prove that there must be a dictator
– Finally, the dictator’s opinion might be very far
from the optimal classification
Introduction
Motivation
Model
Results
Proof (1)
Construction:
We have X={a,b}, and 3 classifiers as follows
The dataset contains two types of agents, with
samples distributed unevenly over a and b
We do not set the labels.
Instead, we denote by Y
all the possible labelings
of an agent’s dataset.
Introduction
Motivation
Model
Results
Proof (2)
Let P be the set of all 6 orders over C
A voting rule is a function of the form f: Pn  C
But our mechanism is a function M: Yn  C !
(its input are labels and not orders)
Lemma 1: there is a valid mapping g: Pn  Yn, s.t.
(M*g) is a voting rule
Introduction
Motivation
Model
Results
Proof (3)
Lemma 2: If M is SP, and guarantees any bounded
approximation ratio, then f=M*g is dictatorial
Proof: (f is onto) any profile that c classifies perfectly
must induce the selection of c
(f is SP) suppose there is a manipulation
By mapping this profile to labels with g, we find a
manipulation of M, in contradiction to its SP
From the G-S theorem, f must be dictatorial
Introduction
Motivation
Model
Results
Proof (4)
Finally, f (and thus M) can only be dictatorial.
We assume w.l.o.g. that the dictator is agent 1 of
type Ia. We now label the data points as follows:
The optimal classifier is cab, which makes 2 errors
The dictator selects ca, which makes m/2 errors
Introduction
Motivation
Model
Results
Real concept classes
• We managed to show that there are no good
(deterministic) SP mechanisms, but only for a
synthetically constructed class.
• We are interested in more common classes, that
are really used in machine learning. For example:
• Linear Classifiers
• Boolean Conjunctions
Introduction
Motivation
Model
Results
Linear classifiers
Only 2
errors
“a”
“b”
cb
ca
cab
Ω(√m) errors
Introduction
Motivation
Model
Results
A lower bound for randomized
SP mechanisms
• A lottery over dictatorships is still bad
– Ω(k) instead of Ω(m), where k is the size of the
largest dataset controlled by an agent ( m ≈ k*n )
• However, it is not clear how to eliminate other
mechanisms
– G-S works only for deterministic mechanisms
– Another theorem by Gibbard [’79] can help
• But only under additional assumptions
Introduction
Motivation
Model
Results
Upper bounds
• So, our lower bounds do not leave much hope
for good SP mechanisms
• We would still like to know if they are tight
A deterministic SP O(m)-approximation is easy:
– break ties iteratively according to dictators
What about randomized SP O(k) mechanisms?
Introduction
Motivation
Model
Results
The iterative random dictator (IRD)
(example with linear classifiers on R1)
v
v
Introduction
Motivation
Model
Results
The iterative random dictator (IRD)
(example with linear classifiers on R1)
v
Iteration 1: 2 errors
v
Introduction
Motivation
Model
Results
The iterative random dictator (IRD)
(example with linear classifiers on R1)
v
Iteration 1: 2 errors
Iteration 2: 5 errors
Iteration 3: 0 errors
v
Introduction
Motivation
Model
Results
The iterative random dictator (IRD)
(example with linear classifiers on R1)
v
Iteration 1:
Iteration 2:
Iteration 3:
Iteration 4:
2 errors
5 errors
0 errors
0 errors
v
Introduction
Motivation
Model
Results
The iterative random dictator (IRD)
(example with linear classifiers on R1)
v
Iteration 1:
Iteration 2:
Iteration 3:
Iteration 4:
Iteration 5:
2 errors
5 errors
0 errors
0 errors
1 error
Theorem: The IRD is O(k2) approximating
for Linear Classifiers in R1
v
Introduction
Motivation
Model
Results
Future work
• Other concept classes
• Other loss functions
• Alternative assumptions on structure of data
• Other models of strategic behavior
• …